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Complex Numbers

A complex number is expressed in the standard form a + bi, where a and b are real numbers and i is defined by i^2 = -1 (that is, i is the square root of -1). For example, 3 + 2i is a complex number.

The bi term is often referred to as an imaginary number (though this may be misleading, as it is no more "imaginary" than the symbolic abstractions we know as the "real" numbers). Thus, every complex number has a real part, a, and an imaginary part, bi.

Complex numbers are often represented on a graph known as the "complex plane," where the horizontal axis represents the infinity of real numbers, and the vertical axis represents the infinity of imaginary numbers. Thus, each complex number has a unique representation on the complex plane: some closer to real; others, more imaginary. If a = b, the number is equal parts real and imaginary.

Very simple transformations applied to numbers in the complex plane can lead to fractal structures of enormous intricacy and astonishing beauty.

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October 11, 2019 at 12:01am
October 11, 2019 at 12:01am
So, speaking of drinking...


How Prohibition Tossed a Wet Blanket on America’s Inventors
New research reveals the link between bars and new inventions.

There have been dark times indeed in the US. But the darkest, most uncivilized period of the 20th century was certainly Prohibition - with McCarthyism running a close second.

I've known this on an instinctual level all my life; after all, my country pays lip service to "freedom" without actually allowing freedom, but Prohibition was the most egregious example of this.

We also pride ourselves, sometimes to an unreasonable extreme, at innovation. So it shouldn't be surprising that booze - specifically, the public consumption thereof - is linked to innovation.

“Researchers tend to think that informal social interactions—people bumping into one another and swapping ideas—is vital for innovation,” Andrews says... “If you press economists on this when they’re giving talks, and ask why it matters that everyone’s in the same city or within a few blocks, they’ll say something like, ‘People get together and talk at the bar,’” says Andrews. “I’ve actually heard this multiple times. [But] I don’t think direct evidence of that has ever existed before.”

Hypothesis: Bars are essential for innovation. Here comes the science.

He downloaded patent data, compared the number granted to inventors in the wet and dry counties before and after statewide prohibition began, and came up with a measurement of the importance of slightly drunken discussion to invention.

The result? A 15 percent decrease in the number of patents. The areas whose saloons shuttered had become less inventive.

Dark. Times.

Prohibition disrupted much more than nightlife. In a tradition dating back to the country’s origins, Americans had long favored beer and bars over coffee and cafés. “The American Revolution was basically plotted in local taverns in Boston and Philadelphia,” says Andrews. Repudiating the tea-loving Brits they’d fought against, Americans viewed bars as venues of culture.

And rightly so.

For his part, Andrews came up with the idea for his research while chatting with other graduate students at a bar, Joe’s Place, in Iowa City. “I find the bar to be a great place to air out lots of ‘bad ideas’ in a setting that is safer than the seminar room,” he says

So, again, it's not drinking but social drinking that's important. Being around people tends to curb the worst effects of overindulgence, provided you're among people who are looking out for you. And presumably, it doesn't matter if an individual drinks or not; what matters is the free exchange of ideas - yet another miracle brought to you by the wonder compound, ethanol.

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